A multigrid solver for incompressible Navier-Stokes on non-staggered grids
Courant Institute, New York University, New York, NY 10012, USA
Uri M. Ascher
Department of Computer Science, University of British Columbia,
Vancouver, BC, V6T 1Z2, Canada
The goal of this work is to develop an efficient multigrid solver for the
steady-state incompressible Navier-Stokes equations on non-staggered grids.
The pressure Poisson equation (PPE) is used instead of the continuity equation
in order to avoid odd-even pressure instability. The differential order of
the resulting system of equations is higher than that of the original system,
so additional boundary conditions are needed. For this, Neumann-type boundary
conditions for the pressure can be derived from the given boundary conditions
(sufficient for the original formulation) using the momentum and continuity
The main achievements of this work are:
The resulting fast solver is capable of producing second order accurate
solutions for the entire range of Reynolds number.
A method of discretizing the Neumann-type boundary conditions for pressure is
developed using a finite volume approach. A relaxation scheme capable of
treating efficiently the resulting difference equations is constructed. The
speed of convergence of the resulting FMG multigrid solver for low Reynolds
numbers is comparable to that for the Poisson equation with Dirichlet boundary
A compact, second order accurate, difference scheme approximating the momentum
equations for high Reynolds numbers is developed.